Formulations of General Relativity Gravity, Spinors and Differential Forms Kirill Krasnov To the memory of two Peter Ivanovichs in my life: Peter Ivanovich Fomin, who got me interested in gravity, and Peter Ivanovich Holod, who formed my taste for mathematics. Contents Preface page x 0 Introduction 1 1 Aspects of Differential Geometry 9 1.1 Manifolds 9 1.2 Differential Forms 13 1.3 Integration of differential forms 19 1.4 Vector fields 22 1.5 Tensors 27 1.6 Lie derivative 30 1.7 Integrability conditions 35 1.8 The metric 36 1.9 Lie groups and Lie algebras 40 1.10 Cartan’s isomorphisms 52 1.11 Fibre bundles 54 1.12 Principal bundles 58 1.13 Hopf fibration 65 1.14 Vector bundles 69 1.15 Riemannian Geometry 75 1.16 Spinors and Differential Forms 77 2 Metric and Related Formulations 83 2.1 Einstein-Hilbert metric formulation 83 2.2 Gamma-Gamma formulation 85 2.3 Linearisation 89 2.4 First order Palatini formulation 92 2.5 Eddington-Schrodinger¨ affine formulation 93 2.6 Unification: Kaluza-Klein theory 94 viii Contents 3 Cartan’s Tetrad Formulation 95 3.1 Tetrad, Spin connection 97 3.2 Einstein-Cartan first order formulation 110 3.3 Teleparallel formulation 112 3.4 Pure connection formulation 114 3.5 MacDowell-Mansouri formulation 116 3.6 Dimensional reduction 119 3.7 BF formulation 121 4 General Relativity in 2+1 Dimensions 133 4.1 Einstein-Cartan and Chern-Simons formulations 133 4.2 The pure connection formulation 137 5 The ”Chiral” Formulation of General Relativity 140 5.1 Hodge star and self-duality in four dimensions 141 5.2 Decomposition of the Riemann curvature 141 5.3 Chiral version of Cartan’s theory 145 5.4 Hodge star and the metric 149 5.5 The ”Lorentz” groups in four dimensions 160 5.6 The self-dual part of the spin connection 169 5.7 The chiral soldering form 172 5.8 Plebanski´ formulation of GR 181 5.9 Linearisation of the Plebanski action 184 5.10 Coupling to matter 190 5.11 Historical remarks 192 5.12 Alternative descriptions related to Plebanski formalism 194 5.13 A second-order formulation based on the 2-form field 198 6 Chiral Pure Connection Formulation 202 6.1 Chiral pure connection formalism for GR 202 6.2 Example: Page metric 222 6.3 Pure connection description of gravitational instantons 229 6.4 First order chiral connection formalism 235 6.5 Example: Bianchi I connections 236 6.6 Spherically symmetric case 244 6.7 Bianchi IX and reality conditions 250 6.8 Connection description of Ricci flat metrics 254 6.9 Chiral pure connection perturbation theory 261 7 Deformations of General Relativity 263 7.1 A natural modified theory 263 Contents ix 8 Perturbative Descriptions of Gravity 268 8.1 Spinor formalism 270 8.2 Spinors and differential operators 275 8.3 Minkowski space metric perturbation theory 287 8.4 Chiral Yang-Mills perturbation theory 289 8.5 Minkowski space chiral first order perturbation theory 294 8.6 Chiral connection perturbation theory 310 9 Higher-Dimensional Descriptions 319 9.1 Twistor space 321 9.2 Euclidean twistors 333 9.3 Quaternionic Hopf Fibration 344 9.4 Twistor description of gravitational instantons 351 9.5 Geometry of 3-forms in seven dimensions 354 7 9.6 G2-structures on S 360 9.7 3-Form version of the twistor construction 373 10 Concluding Remarks 378 Notes 383 References 384 Index 387 Preface Give thanks to God, who made necessary things simple, and complicated things unnecessary. Gregory Skovoroda, Ukrainian Thinker, 1722-1794 There is always another way to say the same thing that doesn’t look at all like the way you said it before. I don’t know what the reason for this is. I think it is somehow a representation of the simplicity of nature? Perhaps a thing is simple if you can describe it fully in several different ways without immediately knowing that you are describing the same thing. Richard Feynman, Nobel Lecture, 1965 Theories of the known, which are described by different physical ideas may be equivalent in all their predictions and are hence scientifically indistinguishable. However, they are not psychologically identical when trying to move from that base into the unknown. For different views suggest different kinds of modifications which might be made and hence are not equivalent in the hypotheses one generates from them in ones attempt to understand what is not yet understood. I, therefore, think that a good theoretical physicist today might find it useful to have a wide range of physical viewpoints and mathematical expressions of the same theory available to him. Richard Feynman, Nobel Lecture, 1965 Formulations of General Relativity. Facing this title the prospective reader should be thinking: What is there to formulate General Relativity? GR can be formulated in one sentence: GR action functional is the integral of the scalar curvature over the manifold. Everything else that is there to say about GR is the consequence of the Preface xi Euler-Lagrange equations one obtains by extremising this action, together with the action for matter fields. How can there be a book about ”formulations”? And why plural? Is not there just the usual Einstein-Hilbert formulation as stated above? A more sophisticated reader will know that there are several equivalent formula- tions of General Relativity. There is the usual metric formulation, and then there is an equivalent formulation in terms of tetrads. But this is all well-known. General Relativity is about physical consequences of the dynamical postulate that fixes the theory. There may be several equivalent ways to define the dynamics. But this does not change the physics. So, one formulation is sufficient to unravel all the physics predicted by the theory. The usual metric formulation is by far the most studied and best understood. Why bother about developing any other equivalent formulation? And then why write a book about such unnecessary alternatives? This is when the above two quotes from the Richard Feynman Nobel lecture become relevant. The first is about an empirical observation that theories that are relevant for describing the world around us tend to admit many different equivalent, but not obviously so, reformulations. The example Feynman has in mind is classical electrodynamics, not gravity. Feynman also notices that there is a deep link between the ”simplicity” of a theory, and the availability of many different not manifestly equivalent descriptions. He goes further and proposes this as the criterion of sim- plicity. This suggests that one can never fully appreciate the simplicity and beauty of General Relativity without absorbing all the different available formulations of this theory. The second quote is a different, but not unrelated thought. There may be equiv- alent formulations of a theory, all leading to the same physical predictions. But such reformulations may be inequivalent if one decides to generalise. The example of most relevance for Feynman is the Hamiltonian and Lagrangian description of classical mechanics. The quantum generalisation of the Hamiltonian description leads to the usual operator formalism for quantum theory. The generalisation of the Lagrangian description leads to path integrals, which is arguably one of Feyn- man’s main contributions to physics. These two equivalent formulations of classical mechanics are certainly not equivalent in terms of the new structures that can be generated from them. The same may well apply to gravity. We do not yet know which of the many available formulations of gravity will lead to the next big step in the quest for understanding the world around us. So, the purpose of this book is to describe all the ”equivalent” formulations of General Relativity that are known to the author, and that also put the geometry of differential forms and fibre bundles at the forefront of the description of gravity. What is meant by a ”formulation” here is a Lagrangian description, in which the dynamical equations are obtained by extremising the corresponding action. This gives the most economic way of defining the theory. xii Preface Some of these equivalent formulations will likely be known to many readers. In particular, this is the already mentioned formulation in terms of tetrads. If this was the complete list, there would be no good reason to write this book. What is known much less, and what really motivated this author to embark on the present project, is that there are some special features of General Relativity in four spacetime dimensions. These special features are related to coincidences that occur precisely in four dimensions. Thus, in any dimension the Riemann curvature can be viewed as a matrix mapping anti-symmetric rank two tensors again into such tensors. And in four dimensions one also has the Hodge star operator that maps anti-symmetric rank two tensors into such tensors. One can ask how these two operations are related or compatible. It is then a simple to check but deep fact that a metric is Einstein if and only if these two operations commute. This fact leads to a whole series of chiral formulations of four dimensional General Relativity that have no analogs in higher spacetime dimensions. It is the development of these formulations, and contrasting them with the more known ones, that will occupy us for the large part of this book. There is no coherent account of these developments in the literature, certainly not in any book on General Relativity. It is our desire to make such a coherent account available that was one of the main motivations for writing this monograph. Another motivation for writing this exposition was our desire to promote the formalism(s) for GR that place the differential forms rather than metrics at the forefront.